ceph/src/boost/boost/math/special_functions/detail/bessel_k1.hpp

   1 //  Copyright (c) 2006 Xiaogang Zhang
   2 //  Copyright (c) 2017 John Maddock
   3 //  Use, modification and distribution are subject to the
   4 //  Boost Software License, Version 1.0. (See accompanying file
   5 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
   6
   7 #ifndef BOOST_MATH_BESSEL_K1_HPP
   8 #define BOOST_MATH_BESSEL_K1_HPP
   9
  10 #ifdef _MSC_VER
  11 #pragma once
  12 #pragma warning(push)
  13 #pragma warning(disable:4702) // Unreachable code (release mode only warning)
  14 #endif
  15
  16 #include <boost/math/tools/rational.hpp>
  17 #include <boost/math/tools/big_constant.hpp>
  18 #include <boost/math/policies/error_handling.hpp>
  19 #include <boost/assert.hpp>
  20
  21 // Modified Bessel function of the second kind of order zero
  22 // minimax rational approximations on intervals, see
  23 // Russon and Blair, Chalk River Report AECL-3461, 1969,
  24 // as revised by Pavel Holoborodko in "Rational Approximations
  25 // for the Modified Bessel Function of the Second Kind - K0(x)
  26 // for Computations with Double Precision", see
  27 // http://www.advanpix.com/2016/01/05/rational-approximations-for-the-modified-bessel-function-of-the-second-kind-k1-for-computations-with-double-precision/
  28 //
  29 // The actual coefficients used are our own derivation (by JM)
  30 // since we extend to both greater and lesser precision than the
  31 // references above.  We can also improve performance WRT to
  32 // Holoborodko without loss of precision.
  33
  34 namespace boost { namespace math { namespace detail{
  35
  36    template <typename T>
  37    T bessel_k1(const T& x);
  38
  39    template <class T, class tag>
  40    struct bessel_k1_initializer
  41    {
  42       struct init
  43       {
  44          init()
  45          {
  46             do_init(tag());
  47          }
  48          static void do_init(const mpl::int_<113>&)
  49          {
  50             bessel_k1(T(0.5));
  51             bessel_k1(T(2));
  52             bessel_k1(T(6));
  53          }
  54          static void do_init(const mpl::int_<64>&)
  55          {
  56             bessel_k1(T(0.5));
  57             bessel_k1(T(6));
  58          }
  59          template <class U>
  60          static void do_init(const U&) {}
  61          void force_instantiate()const {}
  62       };
  63       static const init initializer;
  64       static void force_instantiate()
  65       {
  66          initializer.force_instantiate();
  67       }
  68    };
  69
  70    template <class T, class tag>
  71    const typename bessel_k1_initializer<T, tag>::init bessel_k1_initializer<T, tag>::initializer;
  72
  73
  74    template <typename T, int N>
  75    inline T bessel_k1_imp(const T& x, const mpl::int_<N>&)
  76    {
  77       BOOST_ASSERT(0);
  78       return 0;
  79    }
  80
  81    template <typename T>
  82    T bessel_k1_imp(const T& x, const mpl::int_<24>&)
  83    {
  84       BOOST_MATH_STD_USING
  85       if(x <= 1)
  86       {
  87          // Maximum Deviation Found:                     3.090e-12
  88          // Expected Error Term : -3.053e-12
  89          // Maximum Relative Change in Control Points : 4.927e-02
  90          // Max Error found at float precision = Poly : 7.918347e-10
  91          static const T Y = 8.695471287e-02f;
  92          static const T P[] =
  93          {
  94             -3.621379531e-03f,
  95             7.131781976e-03f,
  96             -1.535278300e-05f
  97          };
  98          static const T Q[] =
  99          {
 100             1.000000000e+00f,
 101             -5.173102701e-02f,
 102             9.203530671e-04f
 103          };
 104
 105          T a = x * x / 4;
 106          a = ((tools::evaluate_rational(P, Q, a) + Y) * a * a + a / 2 + 1) * x / 2;
 107
 108          // Maximum Deviation Found:                     3.556e-08
 109          // Expected Error Term : -3.541e-08
 110          // Maximum Relative Change in Control Points : 8.203e-02
 111          static const T P2[] =
 112          {
 113             -3.079657469e-01f,
 114             -8.537108913e-02f,
 115             -4.640275408e-03f,
 116             -1.156442414e-04f
 117          };
 118
 119          return tools::evaluate_polynomial(P2, T(x * x)) * x + 1 / x + log(x) * a;
 120       }
 121       else
 122       {
 123          // Maximum Deviation Found:                     3.369e-08
 124          // Expected Error Term : -3.227e-08
 125          // Maximum Relative Change in Control Points : 9.917e-02
 126          // Max Error found at float precision = Poly : 6.084411e-08
 127          static const T Y = 1.450342178f;
 128          static const T P[] =
 129          {
 130             -1.970280088e-01f,
 131             2.188747807e-02f,
 132             7.270394756e-01f,
 133             2.490678196e-01f
 134          };
 135          static const T Q[] =
 136          {
 137             1.000000000e+00f,
 138             2.274292882e+00f,
 139             9.904984851e-01f,
 140             4.585534549e-02f
 141          };
 142          if(x < tools::log_max_value<T>())
 143             return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
 144          else
 145          {
 146             T ex = exp(-x / 2);
 147             return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex;
 148          }
 149       }
 150    }
 151
 152    template <typename T>
 153    T bessel_k1_imp(const T& x, const mpl::int_<53>&)
 154    {
 155       BOOST_MATH_STD_USING
 156       if(x <= 1)
 157       {
 158          // Maximum Deviation Found:                     1.922e-17
 159          // Expected Error Term : 1.921e-17
 160          // Maximum Relative Change in Control Points : 5.287e-03
 161          // Max Error found at double precision = Poly : 2.004747e-17
 162          static const T Y = 8.69547128677368164e-02f;
 163          static const T P[] =
 164          {
 165             -3.62137953440350228e-03,
 166             7.11842087490330300e-03,
 167             1.00302560256614306e-05,
 168             1.77231085381040811e-06
 169          };
 170          static const T Q[] =
 171          {
 172             1.00000000000000000e+00,
 173             -4.80414794429043831e-02,
 174             9.85972641934416525e-04,
 175             -8.91196859397070326e-06
 176          };
 177
 178          T a = x * x / 4;
 179          a = ((tools::evaluate_rational(P, Q, a) + Y) * a * a + a / 2 + 1) * x / 2;
 180
 181          // Maximum Deviation Found:                     4.053e-17
 182          // Expected Error Term : -4.053e-17
 183          // Maximum Relative Change in Control Points : 3.103e-04
 184          // Max Error found at double precision = Poly : 1.246698e-16
 185
 186          static const T P2[] =
 187          {
 188             -3.07965757829206184e-01,
 189             -7.80929703673074907e-02,
 190             -2.70619343754051620e-03,
 191             -2.49549522229072008e-05
 192          };
 193          static const T Q2[] =
 194          {
 195             1.00000000000000000e+00,
 196             -2.36316836412163098e-02,
 197             2.64524577525962719e-04,
 198             -1.49749618004162787e-06
 199          };
 200
 201          return tools::evaluate_rational(P2, Q2, T(x * x)) * x + 1 / x + log(x) * a;
 202       }
 203       else
 204       {
 205          // Maximum Deviation Found:                     8.883e-17
 206          // Expected Error Term : -1.641e-17
 207          // Maximum Relative Change in Control Points : 2.786e-01
 208          // Max Error found at double precision = Poly : 1.258798e-16
 209
 210          static const T Y = 1.45034217834472656f;
 211          static const T P[] =
 212          {
 213             -1.97028041029226295e-01,
 214             -2.32408961548087617e+00,
 215             -7.98269784507699938e+00,
 216             -2.39968410774221632e+00,
 217             3.28314043780858713e+01,
 218             5.67713761158496058e+01,
 219             3.30907788466509823e+01,
 220             6.62582288933739787e+00,
 221             3.08851840645286691e-01
 222          };
 223          static const T Q[] =
 224          {
 225             1.00000000000000000e+00,
 226             1.41811409298826118e+01,
 227             7.35979466317556420e+01,
 228             1.77821793937080859e+02,
 229             2.11014501598705982e+02,
 230             1.19425262951064454e+02,
 231             2.88448064302447607e+01,
 232             2.27912927104139732e+00,
 233             2.50358186953478678e-02
 234          };
 235          if(x < tools::log_max_value<T>())
 236             return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
 237          else
 238          {
 239             T ex = exp(-x / 2);
 240             return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex;
 241          }
 242       }
 243    }
 244
 245    template <typename T>
 246    T bessel_k1_imp(const T& x, const mpl::int_<64>&)
 247    {
 248       BOOST_MATH_STD_USING
 249       if(x <= 1)
 250       {
 251          // Maximum Deviation Found:                     5.549e-23
 252          // Expected Error Term : -5.548e-23
 253          // Maximum Relative Change in Control Points : 2.002e-03
 254          // Max Error found at float80 precision = Poly : 9.352785e-22
 255          static const T Y = 8.695471286773681640625e-02f;
 256          static const T P[] =
 257          {
 258             BOOST_MATH_BIG_CONSTANT(T, 64, -3.621379534403483072861e-03),
 259             BOOST_MATH_BIG_CONSTANT(T, 64, 7.102135866103952705932e-03),
 260             BOOST_MATH_BIG_CONSTANT(T, 64, 4.167545240236717601167e-05),
 261             BOOST_MATH_BIG_CONSTANT(T, 64, 2.537484002571894870830e-06),
 262             BOOST_MATH_BIG_CONSTANT(T, 64, 6.603228256820000135990e-09)
 263          };
 264          static const T Q[] =
 265          {
 266             BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00),
 267             BOOST_MATH_BIG_CONSTANT(T, 64, -4.354457194045068370363e-02),
 268             BOOST_MATH_BIG_CONSTANT(T, 64, 8.709137201220209072820e-04),
 269             BOOST_MATH_BIG_CONSTANT(T, 64, -9.676151796359590545143e-06),
 270             BOOST_MATH_BIG_CONSTANT(T, 64, 5.162715192766245311659e-08)
 271          };
 272
 273          T a = x * x / 4;
 274          a = ((tools::evaluate_rational(P, Q, a) + Y) * a * a + a / 2 + 1) * x / 2;
 275
 276          // Maximum Deviation Found:                     1.995e-23
 277          // Expected Error Term : 1.995e-23
 278          // Maximum Relative Change in Control Points : 8.174e-04
 279          // Max Error found at float80 precision = Poly : 4.137325e-20
 280          static const T P2[] =
 281          {
 282             BOOST_MATH_BIG_CONSTANT(T, 64, -3.079657578292062244054e-01),
 283             BOOST_MATH_BIG_CONSTANT(T, 64, -7.963049154965966503231e-02),
 284             BOOST_MATH_BIG_CONSTANT(T, 64, -3.103277523735639924895e-03),
 285             BOOST_MATH_BIG_CONSTANT(T, 64, -4.023052834702215699504e-05),
 286             BOOST_MATH_BIG_CONSTANT(T, 64, -1.719459155018493821839e-07)
 287          };
 288          static const T Q2[] =
 289          {
 290             BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00),
 291             BOOST_MATH_BIG_CONSTANT(T, 64, -1.863917670410152669768e-02),
 292             BOOST_MATH_BIG_CONSTANT(T, 64, 1.699367098849735298090e-04),
 293             BOOST_MATH_BIG_CONSTANT(T, 64, -9.309358790546076298429e-07),
 294             BOOST_MATH_BIG_CONSTANT(T, 64, 2.708893480271612711933e-09)
 295          };
 296
 297          return tools::evaluate_rational(P2, Q2, T(x * x)) * x + 1 / x + log(x) * a;
 298       }
 299       else
 300       {
 301          // Maximum Deviation Found:                     9.785e-20
 302          // Expected Error Term : -3.302e-21
 303          // Maximum Relative Change in Control Points : 3.432e-01
 304          // Max Error found at float80 precision = Poly : 1.083755e-19
 305          static const T Y = 1.450342178344726562500e+00f;
 306          static const T P[] =
 307          {
 308             BOOST_MATH_BIG_CONSTANT(T, 64, -1.970280410292263112917e-01),
 309             BOOST_MATH_BIG_CONSTANT(T, 64, -4.058564803062959169322e+00),
 310             BOOST_MATH_BIG_CONSTANT(T, 64, -3.036658174194917777473e+01),
 311             BOOST_MATH_BIG_CONSTANT(T, 64, -9.576825392332820142173e+01),
 312             BOOST_MATH_BIG_CONSTANT(T, 64, -6.706969489248020941949e+01),
 313             BOOST_MATH_BIG_CONSTANT(T, 64, 3.264572499406168221382e+02),
 314             BOOST_MATH_BIG_CONSTANT(T, 64, 8.584972047303151034100e+02),
 315             BOOST_MATH_BIG_CONSTANT(T, 64, 8.422082733280017909550e+02),
 316             BOOST_MATH_BIG_CONSTANT(T, 64, 3.738005441471368178383e+02),
 317             BOOST_MATH_BIG_CONSTANT(T, 64, 7.016938390144121276609e+01),
 318             BOOST_MATH_BIG_CONSTANT(T, 64, 4.319614662598089438939e+00),
 319             BOOST_MATH_BIG_CONSTANT(T, 64, 3.710715864316521856193e-02)
 320          };
 321          static const T Q[] =
 322          {
 323             BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00),
 324             BOOST_MATH_BIG_CONSTANT(T, 64, 2.298433045824439052398e+01),
 325             BOOST_MATH_BIG_CONSTANT(T, 64, 2.082047745067709230037e+02),
 326             BOOST_MATH_BIG_CONSTANT(T, 64, 9.662367854250262046592e+02),
 327             BOOST_MATH_BIG_CONSTANT(T, 64, 2.504148628460454004686e+03),
 328             BOOST_MATH_BIG_CONSTANT(T, 64, 3.712730364911389908905e+03),
 329             BOOST_MATH_BIG_CONSTANT(T, 64, 3.108002081150068641112e+03),
 330             BOOST_MATH_BIG_CONSTANT(T, 64, 1.400149940532448553143e+03),
 331             BOOST_MATH_BIG_CONSTANT(T, 64, 3.083303048095846226299e+02),
 332             BOOST_MATH_BIG_CONSTANT(T, 64, 2.748706060530351833346e+01),
 333             BOOST_MATH_BIG_CONSTANT(T, 64, 6.321900849331506946977e-01),
 334          };
 335          if(x < tools::log_max_value<T>())
 336             return ((tools::evaluate_polynomial(P, T(1 / x)) / tools::evaluate_polynomial(Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
 337          else
 338          {
 339             T ex = exp(-x / 2);
 340             return ((tools::evaluate_polynomial(P, T(1 / x)) / tools::evaluate_polynomial(Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex;
 341          }
 342       }
 343    }
 344
 345    template <typename T>
 346    T bessel_k1_imp(const T& x, const mpl::int_<113>&)
 347    {
 348       BOOST_MATH_STD_USING
 349       if(x <= 1)
 350       {
 351          // Maximum Deviation Found:                     7.120e-35
 352          // Expected Error Term : -7.119e-35
 353          // Maximum Relative Change in Control Points : 1.207e-03
 354          // Max Error found at float128 precision = Poly : 7.143688e-35
 355          static const T Y = 8.695471286773681640625000000000000000e-02f;
 356          static const T P[] =
 357          {
 358             BOOST_MATH_BIG_CONSTANT(T, 113, -3.621379534403483072916666666666595475e-03),
 359             BOOST_MATH_BIG_CONSTANT(T, 113, 7.074117676930975433219826471336547627e-03),
 360             BOOST_MATH_BIG_CONSTANT(T, 113, 9.631337631362776369069668419033041661e-05),
 361             BOOST_MATH_BIG_CONSTANT(T, 113, 3.468935967870048731821071646104412775e-06),
 362             BOOST_MATH_BIG_CONSTANT(T, 113, 2.956705020559599861444492614737168261e-08),
 363             BOOST_MATH_BIG_CONSTANT(T, 113, 2.347140307321161346703214099534250263e-10),
 364             BOOST_MATH_BIG_CONSTANT(T, 113, 5.569608494081482873946791086435679661e-13)
 365          };
 366          static const T Q[] =
 367          {
 368             BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00),
 369             BOOST_MATH_BIG_CONSTANT(T, 113, -3.580768910152105375615558920428350204e-02),
 370             BOOST_MATH_BIG_CONSTANT(T, 113, 6.197467671701485365363068445534557369e-04),
 371             BOOST_MATH_BIG_CONSTANT(T, 113, -6.707466533308630411966030561446666237e-06),
 372             BOOST_MATH_BIG_CONSTANT(T, 113, 4.846687802282250112624373388491123527e-08),
 373             BOOST_MATH_BIG_CONSTANT(T, 113, -2.248493131151981569517383040323900343e-10),
 374             BOOST_MATH_BIG_CONSTANT(T, 113, 5.319279786372775264555728921709381080e-13)
 375          };
 376
 377          T a = x * x / 4;
 378          a = ((tools::evaluate_rational(P, Q, a) + Y) * a * a + a / 2 + 1) * x / 2;
 379
 380          // Maximum Deviation Found:                     4.473e-37
 381          // Expected Error Term : 4.473e-37
 382          // Maximum Relative Change in Control Points : 8.550e-04
 383          // Max Error found at float128 precision = Poly : 8.167701e-35
 384          static const T P2[] =
 385          {
 386             BOOST_MATH_BIG_CONSTANT(T, 113, -3.079657578292062244053600156878870690e-01),
 387             BOOST_MATH_BIG_CONSTANT(T, 113, -8.133183745732467770755578848987414875e-02),
 388             BOOST_MATH_BIG_CONSTANT(T, 113, -3.548968792764174773125420229299431951e-03),
 389             BOOST_MATH_BIG_CONSTANT(T, 113, -5.886125468718182876076972186152445490e-05),
 390             BOOST_MATH_BIG_CONSTANT(T, 113, -4.506712111733707245745396404449639865e-07),
 391             BOOST_MATH_BIG_CONSTANT(T, 113, -1.632502325880313239698965376754406011e-09),
 392             BOOST_MATH_BIG_CONSTANT(T, 113, -2.311973065898784812266544485665624227e-12)
 393          };
 394          static const T Q2[] =
 395          {
 396             BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00),
 397             BOOST_MATH_BIG_CONSTANT(T, 113, -1.311471216733781016657962995723287450e-02),
 398             BOOST_MATH_BIG_CONSTANT(T, 113, 8.571876054797365417068164018709472969e-05),
 399             BOOST_MATH_BIG_CONSTANT(T, 113, -3.630181215268238731442496851497901293e-07),
 400             BOOST_MATH_BIG_CONSTANT(T, 113, 1.070176111227805048604885986867484807e-09),
 401             BOOST_MATH_BIG_CONSTANT(T, 113, -2.129046580769872602793220056461084761e-12),
 402             BOOST_MATH_BIG_CONSTANT(T, 113, 2.294906469421390890762001971790074432e-15)
 403          };
 404
 405          return tools::evaluate_rational(P2, Q2, T(x * x)) * x + 1 / x + log(x) * a;
 406       }
 407       else if(x < 4)
 408       {
 409          // Max error in interpolated form: 5.307e-37
 410          // Max Error found at float128 precision = Poly: 7.087862e-35
 411          static const T Y = 1.5023040771484375f;
 412          static const T P[] =
 413          {
 414             BOOST_MATH_BIG_CONSTANT(T, 113, -2.489899398329369710528254347931380044e-01),
 415             BOOST_MATH_BIG_CONSTANT(T, 113, -6.819080211203854781858815596508456873e+00),
 416             BOOST_MATH_BIG_CONSTANT(T, 113, -7.599915699069767382647695624952723034e+01),
 417             BOOST_MATH_BIG_CONSTANT(T, 113, -4.450211910821295507926582231071300718e+02),
 418             BOOST_MATH_BIG_CONSTANT(T, 113, -1.451374687870925175794150513723956533e+03),
 419             BOOST_MATH_BIG_CONSTANT(T, 113, -2.405805746895098802803503988539098226e+03),
 420             BOOST_MATH_BIG_CONSTANT(T, 113, -5.638808326778389656403861103277220518e+02),
 421             BOOST_MATH_BIG_CONSTANT(T, 113, 5.513958744081268456191778822780865708e+03),
 422             BOOST_MATH_BIG_CONSTANT(T, 113, 1.121301640926540743072258116122834804e+04),
 423             BOOST_MATH_BIG_CONSTANT(T, 113, 1.080094900175649541266613109971296190e+04),
 424             BOOST_MATH_BIG_CONSTANT(T, 113, 5.896531083639613332407534434915552429e+03),
 425             BOOST_MATH_BIG_CONSTANT(T, 113, 1.856602122319645694042555107114028437e+03),
 426             BOOST_MATH_BIG_CONSTANT(T, 113, 3.237121918853145421414003823957537419e+02),
 427             BOOST_MATH_BIG_CONSTANT(T, 113, 2.842072954561323076230238664623893504e+01),
 428             BOOST_MATH_BIG_CONSTANT(T, 113, 1.039705646510167437971862966128055524e+00),
 429             BOOST_MATH_BIG_CONSTANT(T, 113, 1.008418100718254816100425022904039530e-02)
 430          };
 431          static const T Q[] =
 432          {
 433             BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00),
 434             BOOST_MATH_BIG_CONSTANT(T, 113, 2.927456835239137986889227412815459529e+01),
 435             BOOST_MATH_BIG_CONSTANT(T, 113, 3.598985593265577043711382994516531273e+02),
 436             BOOST_MATH_BIG_CONSTANT(T, 113, 2.449897377085510281395819892689690579e+03),
 437             BOOST_MATH_BIG_CONSTANT(T, 113, 1.025555887684561913263090023158085327e+04),
 438             BOOST_MATH_BIG_CONSTANT(T, 113, 2.774140447181062463181892531100679195e+04),
 439             BOOST_MATH_BIG_CONSTANT(T, 113, 4.962055507843204417243602332246120418e+04),
 440             BOOST_MATH_BIG_CONSTANT(T, 113, 5.908269326976180183216954452196772931e+04),
 441             BOOST_MATH_BIG_CONSTANT(T, 113, 4.655160454422016855911700790722577942e+04),
 442             BOOST_MATH_BIG_CONSTANT(T, 113, 2.383586885019548163464418964577684608e+04),
 443             BOOST_MATH_BIG_CONSTANT(T, 113, 7.679920375586960324298491662159976419e+03),
 444             BOOST_MATH_BIG_CONSTANT(T, 113, 1.478586421028842906987799049804565008e+03),
 445             BOOST_MATH_BIG_CONSTANT(T, 113, 1.565384974896746094224942654383537090e+02),
 446             BOOST_MATH_BIG_CONSTANT(T, 113, 7.902617937084010911005732488607114511e+00),
 447             BOOST_MATH_BIG_CONSTANT(T, 113, 1.429293010387921526110949911029094926e-01),
 448             BOOST_MATH_BIG_CONSTANT(T, 113, 3.880342607911083143560111853491047663e-04)
 449          };
 450          return ((tools::evaluate_polynomial(P, T(1 / x)) / tools::evaluate_polynomial(Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
 451       }
 452       else
 453       {
 454          // Maximum Deviation Found:                     4.359e-37
 455          // Expected Error Term : -6.565e-40
 456          // Maximum Relative Change in Control Points : 1.880e-01
 457          // Max Error found at float128 precision = Poly : 2.943572e-35
 458          static const T Y = 1.308816909790039062500000000000000000f;
 459          static const T P[] =
 460          {
 461             BOOST_MATH_BIG_CONSTANT(T, 113, -5.550277247453881129211735759447737350e-02),
 462             BOOST_MATH_BIG_CONSTANT(T, 113, -3.485883080219574328217554864956175929e+00),
 463             BOOST_MATH_BIG_CONSTANT(T, 113, -8.903760658131484239300875153154881958e+01),
 464             BOOST_MATH_BIG_CONSTANT(T, 113, -1.144813672213626237418235110712293337e+03),
 465             BOOST_MATH_BIG_CONSTANT(T, 113, -6.498400501156131446691826557494158173e+03),
 466             BOOST_MATH_BIG_CONSTANT(T, 113, 1.573531831870363502604119835922166116e+04),
 467             BOOST_MATH_BIG_CONSTANT(T, 113, 5.417416550054632009958262596048841154e+05),
 468             BOOST_MATH_BIG_CONSTANT(T, 113, 4.271266450613557412825896604269130661e+06),
 469             BOOST_MATH_BIG_CONSTANT(T, 113, 1.898386013314389952534433455681107783e+07),
 470             BOOST_MATH_BIG_CONSTANT(T, 113, 5.353798784656436259250791761023512750e+07),
 471             BOOST_MATH_BIG_CONSTANT(T, 113, 9.839619195427352438957774052763490067e+07),
 472             BOOST_MATH_BIG_CONSTANT(T, 113, 1.169246368651532232388152442538005637e+08),
 473             BOOST_MATH_BIG_CONSTANT(T, 113, 8.696368884166831199967845883371116431e+07),
 474             BOOST_MATH_BIG_CONSTANT(T, 113, 3.810226630422736458064005843327500169e+07),
 475             BOOST_MATH_BIG_CONSTANT(T, 113, 8.854996610560406127438950635716757614e+06),
 476             BOOST_MATH_BIG_CONSTANT(T, 113, 8.981057433937398731355768088809437625e+05),
 477             BOOST_MATH_BIG_CONSTANT(T, 113, 2.519440069856232098711793483639792952e+04)
 478          };
 479          static const T Q[] =
 480          {
 481             BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00),
 482             BOOST_MATH_BIG_CONSTANT(T, 113, 7.127348248283623146544565916604103560e+01),
 483             BOOST_MATH_BIG_CONSTANT(T, 113, 2.205092684176906740104488180754982065e+03),
 484             BOOST_MATH_BIG_CONSTANT(T, 113, 3.911249195069050636298346469740075758e+04),
 485             BOOST_MATH_BIG_CONSTANT(T, 113, 4.426103406579046249654548481377792614e+05),
 486             BOOST_MATH_BIG_CONSTANT(T, 113, 3.365861555422488771286500241966208541e+06),
 487             BOOST_MATH_BIG_CONSTANT(T, 113, 1.765377714160383676864913709252529840e+07),
 488             BOOST_MATH_BIG_CONSTANT(T, 113, 6.453822726931857253365138260720815246e+07),
 489             BOOST_MATH_BIG_CONSTANT(T, 113, 1.643207885048369990391975749439783892e+08),
 490             BOOST_MATH_BIG_CONSTANT(T, 113, 2.882540678243694621895816336640877878e+08),
 491             BOOST_MATH_BIG_CONSTANT(T, 113, 3.410120808992380266174106812005338148e+08),
 492             BOOST_MATH_BIG_CONSTANT(T, 113, 2.628138016559335882019310900426773027e+08),
 493             BOOST_MATH_BIG_CONSTANT(T, 113, 1.250794693811010646965360198541047961e+08),
 494             BOOST_MATH_BIG_CONSTANT(T, 113, 3.378723408195485594610593014072950078e+07),
 495             BOOST_MATH_BIG_CONSTANT(T, 113, 4.488253856312453816451380319061865560e+06),
 496             BOOST_MATH_BIG_CONSTANT(T, 113, 2.202167197882689873967723350537104582e+05),
 497             BOOST_MATH_BIG_CONSTANT(T, 113, 1.673233230356966539460728211412989843e+03)
 498          };
 499          if(x < tools::log_max_value<T>())
 500             return ((tools::evaluate_polynomial(P, T(1 / x)) / tools::evaluate_polynomial(Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
 501          else
 502          {
 503             T ex = exp(-x / 2);
 504             return ((tools::evaluate_polynomial(P, T(1 / x)) / tools::evaluate_polynomial(Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex;
 505          }
 506       }
 507     }
 508
 509     template <typename T>
 510     T bessel_k1_imp(const T& x, const mpl::int_<0>&)
 511     {
 512        if(boost::math::tools::digits<T>() <= 24)
 513           return bessel_k1_imp(x, mpl::int_<24>());
 514        else if(boost::math::tools::digits<T>() <= 53)
 515           return bessel_k1_imp(x, mpl::int_<53>());
 516        else if(boost::math::tools::digits<T>() <= 64)
 517           return bessel_k1_imp(x, mpl::int_<64>());
 518        else if(boost::math::tools::digits<T>() <= 113)
 519           return bessel_k1_imp(x, mpl::int_<113>());
 520        BOOST_ASSERT(0);
 521        return 0;
 522     }
 523
 524     template <typename T>
 525    inline T bessel_k1(const T& x)
 526    {
 527       typedef mpl::int_<
 528          ((std::numeric_limits<T>::digits == 0) || (std::numeric_limits<T>::radix != 2)) ?
 529          0 :
 530          std::numeric_limits<T>::digits <= 24 ?
 531          24 :
 532          std::numeric_limits<T>::digits <= 53 ?
 533          53 :
 534          std::numeric_limits<T>::digits <= 64 ?
 535          64 :
 536          std::numeric_limits<T>::digits <= 113 ?
 537          113 : -1
 538       > tag_type;
 539
 540       bessel_k1_initializer<T, tag_type>::force_instantiate();
 541       return bessel_k1_imp(x, tag_type());
 542    }
 543
 544 }}} // namespaces
 545
 546 #ifdef _MSC_VER
 547 #pragma warning(pop)
 548 #endif
 549
 550 #endif // BOOST_MATH_BESSEL_K1_HPP
 551